We show that the LSD of the above matrices continue to exist in this dependent situation under appropriate conditions on the spectral density of the process. The LSD turn out to be appropriate
mixtures of, the normal distribution, the “symmetric” square root of the chisquare distribution, and, some other related distributions. Quite expectedly, the spectral density of the process is involved in
these mixtures. Our results also reduce to the results quoted above for the i.i.d. situation.
In Section 2 we describe the nature of the eigenvalues of the above matrices, describe the spectral density and set up notation. In Section 3 we state the main results and report some simulation
which demonstrate our theoretical results. The main proofs are given in Section 4 and the proofs of some auxiliary Lemma are given in the Appendix.
Some of the results reported in this article have been reported in the not to be published technical reports Bose and Saha [2008a], Bose and Saha [2008b], Bose and Saha [2009].
2 Preliminaries
2.1 Spectral density and related facts
Under Assumptions B and C, γ
h
= C ov x
t+h
, x
t
is finite and P
j ∈Z
|γ
j
| ∞. The spectral density function f
of {x
n
} exists, is continuous, and is given by f
ω = 1
2π X
k ∈Z
γ
k
expikω = 1
2π γ
+ 2 X
k ≥1
γ
k
coskω for ω
∈ [0, 2π]. Let
I
n
ω
k
= 1
n
n −1
X
t=
x
t
e
−itω
k
2
, k = 0, 1, . . . , n − 1,
2.1 denote the periodogram of
{x
i
} where ω
k
= 2πkn are the Fourier frequencies. Let C
= {t ∈ [0, 1] : f 2πt = 0} and C
′
= {t ∈ [0, 12] : f 2πt = 0}. 2.2
Define ψe
i ω
=
∞
X
j= −∞
a
j
e
i j ω
, ψ
1
e
i ω
= R[ψe
i ω
], ψ
2
e
i ω
= I [ψe
i ω
], 2.3
where a
i
’s are the moving average coefficients in the definition of x
n
. It is easy to see that |ψe
i ω
|
2
= [ψ
1
e
i ω
]
2
+ [ψ
2
e
i ω
]
2
= 2π f ω. Let
B ω =
ψ
1
e
i ω
−ψ
2
e
i ω
ψ
2
e
i ω
ψ
1
e
i ω
and for g
≥ 2,
B ω
1
, ω
2
, .., ω
g
=
ψ
1
e
i ω
1
−ψ
2
e
i ω
1
· · · ψ
2
e
i ω
1
ψ
1
e
i ω
1
· · · ψ
1
e
i ω
2
−ψ
2
e
i ω
2
· · · ψ
2
e
i ω
2
ψ
1
e
i ω
2
· · · ..
. · · ·
ψ
1
e
i ω
g
−ψ
2
e
i ω
g
· · · ψ
2
e
i ω
g
ψ
1
e
i ω
g
.
2467
The above functions will play a crucial role in the statements and proofs of the main results later.
2.2 Description of eigenvalues
We now describe the eigenvalues of the four classes of matrices. Let [x] be the largest integer less than or equal to x.
i Circulant matrix. Its eigenvalues {λ
i
} are see for example Brockwell and Davis [2002], λ
k
= 1
p n
n −1
X
l=
x
l
e
i ω
k
l
= b
k
+ ic
k
∀ k = 1, 2, · · · , n, where
ω
k
= 2πk
n , b
k
= 1
p n
n −1
X
l=
x
l
cosω
k
l , c
k
= 1
p n
n −1
X
l=
x
l
sinω
k
l .
2.4
ii Symmetric circulant matrix. The eigenvalues
{λ
i
} of SC
n
are given by: a for n odd:
λ =
1 p
n x
+ 2
[n2]
X
j= 1
x
j
λ
k
= 1
p n
x + 2
[n2]
X
j= 1
x
j
cos 2πk j
n , 1
≤ k ≤ [n2] b for n even:
λ =
1 p
n x
+ 2
n 2
−1
X
j= 1
x
j
+ x
n 2
λ
k
= 1
p n
x + 2
n 2
−1
X
j= 1
x
j
cos 2πk j
n + −1
k
x
n 2
, 1 ≤ k ≤
n 2
with λ
n −k
= λ
k
for 1 ≤ k ≤ [n2] in both the cases.
iii Palindromic Toeplitz matrix. As far as we know, there is no formula solution for the eigenval- ues of the palindromic Toeplitz matrix. As pointed out already, since the n
× n principal minor of P T
n+ 1
is SC
n
, by interlacing inequality P T
n
and SC
n
have identical LSD.
iv Reverse circulant matrix. The eigenvalues are given in Bose and Mitra [2002]:
λ
= n
−12
P
n −1
t=
x
t
λ
n 2
= n
−12
P
n −1
t=
−1
t
x
t
, if n is even λ
k
= −λ
n −k
= p
I
n
ω
k
, 1 ≤ k ≤ [
n −1
2
].
v k-circulant matrix. The structure of its eigenvalues is available in Zhou 1996. A more de- tailed analysis and related properties of the eigenvalues, useful in the present context, have been
2468
developed in Section 2 of Bose, Mitra and Sen [2008]. Let ν = ν
n
:= cos2πn + i sin2πn and λ
k
=
n −1
X
l=
x
l
ν
kl
, 0 ≤ j n.
2.5 For any positive integers k, n, let p
1
p
2
. . . p
c
be all their common prime factors so that, n = n
′ c
Y
q= 1
p
β
q
q
and k = k
′ c
Y
q= 1
p
α
q
q
. Here α
q
, β
q
≥ 1 and n
′
, k
′
, p
q
are pairwise relatively prime. For any positive integer s, let Z
s
= {0, 1, 2, . . . , s − 1}. Define the following sets
Sx = {x k
b
mod n
′
: b ≥ 0}, 0 ≤ x n
′
. For any set A, let
|A| denote its cardinality. Let g
x
= |Sx| and υ
k ,n
′
= {x ∈ Z
n
′
: g
x
g
1
} .
2.6 We observe the following about the sets Sx.
1. Sx = {x k
b
mod n
′
: 0 ≤ b |Sx|}.
2. For x 6= u, either Sx = Su or Sx ∩ Su = φ. As a consequence, the distinct sets from the
collection {Sx : 0 ≤ x n
′
} forms a partition of Z
n
′
. We shall call
{Sx} the eigenvalue partition of {0, 1, 2, . . . , n−1} and we will denote the partitioning sets and their sizes by
{P ,
P
1
, . . . , P
l −1
}, and n
i
= |P
i
|, 0 ≤ i l. 2.7
Define y
j
:= Y
t ∈P
j
λ
t y
, j = 0, 1, . . . , l − 1 where y = nn
′
. Then the characteristic polynomial of A
k ,n
whence its eigenvalues follow is given by χ
A
k ,n
= λ
n −n
′
ℓ−1
Y
j=
λ
n
j
− y
j
.
2.8
3 Main results
For any Borel set B, LebB will denote its Lebesgue measure in the appropriate dimension.
2469
3.1 Circulant matrix
